Triangle $$ABC$$ has $$BC=20.$$ The incircle of the triangle evenly trisects the median $$AD.$$ If the area of the triangle is $$m \sqrt{n}$$ where $$m$$ and $$n$$ are integers and $$n$$ is not divisible by the square of a prime, find $$m+n.$$

(第二十三届AIME1 2005 第15题)